Vector-Valued Stochastic Integration With Respect to Semimartingales in the Dual of Nuclear Space

Abstract

In this work, we investigate a theory of stochastic integration for operator-valued processes with respect to semimartingales taking values in the dual of a nuclear space. Our construction of this particular stochastic integral relies on previous results from [Electron. J. Probab., Volume 26, paper no. 147, 2021], together with specific tools which share some common features with good integrators in finite dimensions. We investigate various properties of this stochastic integral together with applications. In particular we obtain approximations by Riemann sums results, and provide an alternative proof of \"{U}st\"{u}nel's version of It\^{o}'s formula involving of distributions.